Don't know about "best", but for reasonably cheap, available and not terribly much effort, Lagostina (the cooking pot folks) sell a Stainless Steel polishing powder that has removed discoloration and other flaws from my stainless steel stovetop covers. I think I got that at Canadian Tire some years ago. It seems to combine chemical and physical polishing. Takes some elbow grease.
Autosol is also available at most stores and does a decent job with stainless stuff as well. More elbow grease as it doesn't have any chemicals to dissolve the discoloration - just abrasives.
If you are near any decent kitchen store, you will find they tend to have all kinds of stuff for polishing stainless. Don't fall for anything in a spray can - those are just cleaner-wax type products that are meant for stainless fridges and such. Get one for pots.
I also got some serious discoloration and goo off a stainless ring on my stove using a copper scrubbing pad (dollar store) but that is only appropriate for dull finishes - it seems to dull the finish of highly polished stainless.
Good luck!